Whats the problem with 24bit

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The thing is that the human eye (a perfect one) can see over a 100 million different colors.

The claim about the 16.7 million colors is correct in the sense that a screen cannot produce the full range of colors the eye can see. It simply doesn't cover a large enough range of the spectrum.

The problem with banding does in fact not come from only 256 shades being available, but because most screens do not remap the intensities in a 1:1 ratio, therefore you get visible color jumps between certain values. This is particularily noticable on LCD screens, as they cover a smaller color range, and have sharper definition compared to CCT screens.

Internally it makes sense to compute colors at a higher quality, to avoid clamping and other things, but when it comes to displaying colors 24bit is enough as long as the display technology is at it's current level.